Evariste
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Why multiplication of 142857 with 2,3,4,5,6 gives the same digits shifted?
7 votes

Well, one can note that $142857$ is a bit of a special number in that $\frac{1}{7}=0.\overline{142857}$, which means that, by long division, one obtains $\overline{142857}=142857142857142857...$ ad ...

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The first $n$ such that $n!$ has more than or equal $10000$ trailing zeros
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7 votes

The highest power of $2$ dividing $n!$ is $\lfloor \frac{n}{2} \rfloor+\lfloor \frac{n}{4} \rfloor+\lfloor \frac{n}{8} \rfloor+...$ The highest power of $5$ dividing $n!$ is $\lfloor \frac{n}{5} \...

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How to evaluate $\sqrt {6\sqrt {6\sqrt{\cdots}}}$
7 votes

Another possible solution: just compute the geometric sum in the exponent. $$x=6^{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}...}=6$$ Could be done in your head if you wanted to.

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Definition and ideals of $\mathbb{Z}/n\mathbb{Z}$
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6 votes

Here's a very simple answer for your second question, containing some essential rules of thumb to have in mind to understand basic ring theory, and in particular that of PIDs such as $\mathbb{Z}$. In ...

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How do I show that a finite group $G$ of order $n$ is cyclic if there is at most one subgroup of order $d$ for each $d\mid n$?
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5 votes

The usual way to go about this classic problem is the following. Let $G=\cup G_d$ where $G_d$ is the set of elements of $G$ of order $d$ for each $d|n$. Since there's at most one subgroup of order $d$,...

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Find the remainder of a number when divided by $9$
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5 votes

Hint : $\sum a_k\times10^k\equiv\sum a_k \pmod {9}$

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How to calculate an $n$ digit decimal approximation of a fraction?
4 votes

A simple algorithm is the following : Take integer part $\rightarrow$ multiply the numerator of the remainder by $10$ (in base $10$) $\rightarrow$ divide by the denominator $\rightarrow$ repeat ...

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How do you evaluate the integral $\int_{0}^{\infty} \frac{x \sin x}{1+x^2} dx$?
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3 votes

Use a contour integral in the upper complex plane if you know residue calculus. $\int_0^{+\infty}\frac{x\sin(x)}{1+x^2}dx=\frac12 \operatorname{Im} \int_{\mathbb{R}}\frac{xe^{ix}}{1+x^2}=\frac 12\...

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FInd the missing digit in $2^{29}$ given all nine digits differ
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3 votes

$2^{29}\equiv2^{-1}\equiv\frac{1}{2}\equiv\frac{10}{2}\equiv5\equiv \text{s} \pmod 9$ where $\text{s}$ denotes the sum of its digits. The sum of the ten digits is $\text{S}=0+1+2+3+4+5+6+7+8+9=\frac{...

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Show that if $a,b \in \mathbb{N}$ have remainders in the set $\{1,4\}$ after division by $5$, then so does their product.
3 votes

$1^2\equiv4^2\equiv1 \pmod 5$ $1\times4\equiv4\times1\equiv4 \pmod 5$ So yes, the remainder of the product is always either $1$ or $4$

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What is $a^{(\log_ab)^2}$?
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3 votes

$$a^{(\log_ab)^2}=a^{\log_a(b)\times \log_a(b)}=b^{\log_a(b)}$$

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Two persons A and B throw a die. Who wins given the below conditions?
3 votes

Draw a tree and notice that, by periodicity, the probability that B wins is : $$P_B=\frac{5}{6}+\frac{1}{6} \times \frac{1}{6} \times(\frac{5}{6}+...)$$ So we can express this probability as the ...

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What is the formula to calculate the number of divisors of $n!$
2 votes

If $p \leq n$ is prime then it has exponent $\lfloor \frac{n}{p} \rfloor+\lfloor \frac{n}{p^2}\rfloor +\cdots+\lfloor \frac{n}{p^\alpha}\rfloor$ in $n!$ where $\alpha$ is the largest integer $\geq 1$ ...

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Brownian motions
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2 votes

If $A_t$ and $B_t$ are independent Brownian motions, then $\sin(a)A_t+\cos(a)B_t$ is a centered Gaussian process (due to independence, $\sin(a)A_t+\cos(a)B_t$ has law $N(0,(\sin(a)^2+\cos(a)^2)t)=N(0,...

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Find the maximum value of $\sqrt{x - 144} + \sqrt{722 - x}$
2 votes

We have that $P^2=578+2\sqrt{(x-144)(722-x)}$ To maximize $P^2$ is to maximize $(x-144)(722-x)$, which is a downward parabola. By symmetry, it is maximized right between the two roots, i.e. at $x=\...

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Show $\int_0^\infty xe^{-ax^2}dx=\frac{1}{2a}$
2 votes

$\int_0^{+\infty}xe^{-ax^2}dx=\frac{1}{-2a}\int_0^{+\infty}-2axe^{-ax^2}dx=\frac{1}{-2a}[e^{-ax^2}]_0^{+\infty}=\frac{1}{2a}$

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Are these two scenarios equivalent ? (random walks on chessboard)
2 votes

Answering self: $\sigma_2>\sigma_1$ The two problems are not equivalent because in random walk $1$, a random path is chosen uniformly among the possible ones, but not in random walk $2$ Indeed, ...

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Remainder in this case using modular arithmetic
2 votes

$65=13\times5$ $870^{479}\equiv0 \pmod{5}$ $\color{red}{87}0^{479}\equiv\color{red}{9}0^{479}\equiv(-1)^{479}\equiv-1\pmod{13}$ So $x=870^{479}=-1+13k\equiv0 \pmod 5 \rightarrow k\equiv\frac{1}{13}\...

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solve x for a cubic congruence equation with large prime mod.
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2 votes

I can't find an easy way to show this by hand (although it is still possible). However, I am presenting an answer assuming the prime factorization of $z$ is known (or that the use of a calculator is ...

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Help proving $odd(m^2+n^2) \implies odd((m+n)^2)$
2 votes

$(m+n)^2=m^2+n^2+2mn\equiv m^2+n^2 \pmod 2$ So $m^2+n^2$ odd implies $(m+n)^2$ odd.

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Prime factors of numbers of form $a^2+2b^2$
2 votes

I think I fixed the proof, and it is rather elementary, but not so simple... We're working with $N>1$ ($N=1=1^2+2\times0^2$ has no inexpressible prime factor) Notice that according to ...

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What are the last two digits of 43^23^33?
2 votes

Let's work $\pmod {100}$ Notice that $43^2=(50-7)^2\equiv7^2\equiv49 \pmod {100}$ and likewise $49^2\equiv1 \pmod {100}$ Therefore, $43^4\equiv 1 \pmod {100}$ $23^{33}\equiv (-1)^{33}\equiv-1 \...

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Double summation to find a closed-form formula
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2 votes

$$\sum_{i=1}^n i3^i=3+2\times3^2+3\times3^3+...+n\times3^n=(3+3^2+...+3^n)+(3^2+3^3+...+3^n)+...+3^n=\sum_{i=1}^n \sum_{k=i}^n3^k=\sum_{i=1}^n\frac{3^i-3^{n+1}}{1-3}=\frac{1}{2}\sum_{i=1}^n(3^{n+1}-3^...

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Proof that $\sum_{n=1}^{\infty}2^{-3n} = \frac{1}{7}$
2 votes

$$\sum_{n=0}^{+\infty}2^{-3n}=\sum_{n=0}^{+\infty}(2^{-3})^n=\sum_{n=0}^{+\infty}(\frac{1}{8})^n=\frac{1}{1-\frac{1}{8}}=\frac{1}{\frac{7}{8}}=\frac{8}{7}$$

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How can I find the last digit of $17^{68}$ and the last both digits of $14^{200}$?
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2 votes

$100=4\times25$ and $gcd(4,25)=1$ $14^{200}\equiv2^{200}\times7^{200}\equiv0$ mod $4$ (since $4=2^2$) $14^{200}\equiv(-11)^{200}\equiv11^{200}\equiv121^{100}\equiv(-4)^{100}\equiv4^{100}\equiv(-1)^{...

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Proabability of drawing a white ball
2 votes

$U=(N_{black},N_{white})$ At the start : $U_1=U_2=U_3=(5,7)$ The probability of picking a random black ball from $U_1$ is $\frac{5}{12}$, the probability of picking a white one is $\frac{7}{12}$ ...

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If Mr. X was born on April 16, 1987 what day is 2016 days after he was born?
2 votes

$2016 = 365 \times 5+191$ So 5 years and 189 days have elapsed (191-2 days because there are two leap year, one every 4 years, except when the year number is divisible by 4, and 100 but not by 400). ...

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Choosing at least 2 women from 7 men and 4 women
1 votes

The number of ways is $$\binom{4}{2}\binom{7}{4}+\binom{4}{3}\binom{7}{3}+\binom{4}{4}\binom{7}{2} =6\times35+4\times35+21 = 371$$ "At least two women" means either $2$, $3$ or $4$.

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Friends and persons
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1 votes

First, pick two friends, this yields $\binom{3}{2}$. Then arrange them, this yields a factor of $2!$. Then, note that the "exactly" means the third not-picked friend should not sit next to ...

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Listing order of all the elements in multiplicative group and all of its generators
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1 votes

These exercises are easy and usually very short. I explain with words how to go around these types of problems. Note however that finding generators is non-trivial in general. Finding generators is ...

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